Network Reliability-Based Optimal Toll Design

Transcription

1 Journal of Advanced Transortation, Vol. 42, No. 3, Network Reliability-Based Otimal Toll Design Hao Li Michiel C.J. Bliemer Piet H.L.Bovy Otimal toll design from a network reliability oint of view is addressed in this aer. Imroving network reliability is roosed as a olicy objective of road ricing. A reliabilitybased otimal toll design model, where on the uer level network erformance including travel time reliability is otimized, while on the lower level a dynamic userequilibrium is achieved, is resented. Road authorities aim to otimize network travel time reliability by setting tolls in a network design roblem. Travele are influenced by these tolls and make route and tri decisions by considering travel times and tolls. Network erformance reliability is analyzed for a degradable network with elastic and fluctuated travel demand, which integrates reliability and uncertainty, dynamic network equilibrium models, and Monte Carlo methods. The roosed model is alied to a small hyothesized network for which otimal tolls are derived. The network travel time reliability is indeed imroved after imlementing otimal tolling system. Tris may have a somewhat higher, but more reliable, travel time. Introduction Road ricing has been advocated as a otentially owerful travel demand management (TDM) strategy caable of significantly influencing travel demand characteristics (Cain et al., 2001; Small et al., 2005). Generally, the olicy objectives of road ricing can be summarized as: managing demand, otimizing congestion level, reducing environmental imacts, maximizing social welfare gains, raising revenues to recou maintenance cost and construction cost, etc. Hao Li, Michiel C.J. Bliemer and Piet H.L.Bovy, Delft Univeity of Technology, Faculty of Civil Engineering and Geosciences, Deartment of Transort and Planning, Delft, The Netherlands Received: October 2007 Acceted: January 2008

2 312 H. Li, M.C.J. Bliemer, P.H.L.Bovy The otimal charges obtained with different olicy objectives are different. Nowadays, reliability is becoming an increasingly imortant erformance characteristic of road networks and transort services. Reliability is generally defined as the ability of a system or rocess to erform a required function under given environmental and oerational conditions and for a stated eriod of time (see Hoyland and Rausand, 1994). With regard to road networks, the oerational conditions could be normal and abnormal. The required function can be described in terms of outut variables like travel time, throughut, accessibility, equity, and so forth (Yin and Ieda, 2001). Reliability, however defined, rovides a measure of the stability of the quality of service, which the transort system can offer to its use. Whereas road ricing has been mainly considered to relieve congestion, it will be argued in this aer that it can also hel rovide more reliable transort services to the use. To our best knowledge, no studies have adoted network travel time reliability as an objective of road ricing to design toll systems. Some researche (e.g., Brownstone and Small, 2005) suggested to evaluate road ricing from a network reliability eective. Some other researche (Chan and Lam, 2005; Li et al., 2006) have investigated the imacts of road ricing on network reliability. This aer rooses an otimal toll design model with network travel time reliability as one of the olicy objectives. Definition of Network Reliability-Based Toll Design Problem In an otimal toll design roblem, different otimal toll schemes will result from different olicy objectives. Nowadays reliability of a road network, the stability of the service the road authority can rovide to its use, is becoming increasingly imortant. Designing toll levels from such a reliability eective is a new roosition of the road ricing objective. Concetually, in the short term, road ricing influences travele tri choices, dearture time choices, mode choices, and route choices. Tri choices and mode shifts together may result in a dro of travel demand during the tolling eriod. Dearture time choices, in which travele deart earlier or later to avoid traveling in the tolled eriod, lead to temoral shifts of travel demand, thus a decrease of travel demand as well within the tolling eriod. Consequently, decreased travel demand

3 Network reliability-based 313 and route flow shifts in the charged eriod are the revailing determinants of network reliability. In this aer, it is assumed that the road authority aims to imrove road network reliability and to otimize system erformance (i.e., imrove reliability and minimize travel time on a network level). The tolls set by the road authority are assumed to influence travele in their route and tri decisions where the decision to make a tri or not during the tolled eriod reresents temoral and modal shifts as well. In addition, the travele are assumed to take only travel time and tolls into account when making these decisions, however not tri reliability. Therefore, in this aer reliability is redominantly viewed as a network objective from a lanning oint of view, however it could be extended towards a user s objective as well (for future work). Network erformance reliability will be analyzed considering stochastic network characteristics (i.e., stochastic link caacities), elastic demand and fluctuated travel demand from day-to-day. A network reliability-based otimal toll design model, by choosing the otimal toll levels for a subset of links, subject to dynamic equilibrium traffic assignment is roosed in this aer. A dynamic travel analysis is followed because it will give more accurate estimates of congestion atterns and derived tri characteristics such as travel times. Due to the integrated stochastic roerties, dynamic assignment, elastic demands, and various toll schemes, this reliabilitybased toll design roblem becomes very comlex, that s why the alication is only done to simlistic networks so far. The otimal toll design roblem is a bi-level roblem. On the uer level, the road authority determines the tolls on a set of links aiming to imrove network travel time reliability and minimize network travel time, while on the lower level, travele react on these tolls and change their route choices or decide to change mode or not travel at all. These latter two decisions are assumed to be simultaneously catured in an elastic demand framework. In terms of traffic assignment, dearture time choice is not modeled since we will only analyze road ricing schemes with uniform tolls during the charging eriod. While in terms of reliability definition, secial attention is needed for the time dimensions (within-day and over days). Reliability is defined by a long term series of day-to-day tri travel times for a articular eriod of the day, caused by day-to-day fluctuations in demand and caacity. In general, reliability is a time-deendent variable within-day, but reflecting day-to-day variations in exerienced travel times. To that eriod of the day, a demand attern encasulated by an analysis eriod (mostly of

4 314 H. Li, M.C.J. Bliemer, P.H.L.Bovy longer duration) is attached. The traffic flows including their tri times are determined for that eriod of the day following from that demand attern. Mathemarical Formulation The roosed design roblem is a mathematical rogram with equilibrium constraints (MPEC) where the uer level is an otimization roblem and the lower level is an equilibrium roblem (Luo et al., 1996; Lawhonganich and Hearn, 2004). MPEC is a secial case of a bi-level rogramming roblem. This section formulates the reliability-based toll design roblem as an MPEC in which the design variables are the toll levels on each of the links in the network. Formulation of Lower Level Equilibrium Problem Consider a network G = ( N, A), where N is the set of nodes, and A is the set of directed links. Let R N denote the set of origins and S N denote the set of destinations in the network, where each combination (,), r R, s S, denotes a certain origin-destination (OD) air in the network. Furthermore, let P denote the set of feasible routes from r to s. Time is considered to be discretized into small time eriods t T. Travel demand is assumed for the eriod K T, also subdivided into small eriods k K. For each link a A the travel time τ a () t on a when entering the link at time t is exressed as: ( ) τ () t = f q, C, (1) a a a where fa () is a link secific function. The vector q denote the inflow rates and outflow rates (veh/h) on link a in all time eriods (revious and current time eriods). C a is the caacity (veh/h) on link a. Reliability of the network is the main focus in our otimal toll design roblem. The sources of network unreliability within the transort network can be classified into two main categories: (a) travel demand variation, and (b) suly variation (e.g. link caacities). Hence, both the link flow rates q

5 Network reliability-based 315 as well as the caacities τ () t is also stochastic. a C a are considered stochastic variables, such that In reality, caacity is not a continuous variable, deendent on the number of lanes. However in order to cature the stochastic roerties of link caacity caused by weather, etc, we assume link caacity as a continuous variable, following a class of distribution. Current research alies different assumtions of the distribution of stochastic link caacity, such as uniform distribution, normal distribution, exonential distribution, gamma distribution, weibull distribution (Brilon, Geistefeldt et al., 2005), etc. Deending on different characteristics of road facilities in different area, different link caacity distributions can be chosen. In our analysis, we assume each link caacity C a following a normal distribution indeendently with ω % coefficient of variation (value to be derived from emirical data; see AVV, 1999), exressed as: 2 (,( ) ) C N C ω C (2) a a a where C a is the mean caacity. The travel demand Q ( k ) (veh/h) for each OD air (,) and dearture eriod k K is assumed to follow a normal distribution indeendently as well with β % coefficient of variation (derived from emirical data in Tu (2008)), exressed as: 2 ( ( ) ) Q ( k) N Q ( k), β Q ( k) (3) where Q ( k ) is the mean elastic demand for time eriod k. It should be noted that this elastic demand is derived from some base travel demand Qo ( k ) and deends on the generalized OD costs c ( k ) ( ) which include the tolls denoted by the vector of (uniform) link tolls θ [ θ a ] ( ). The mean elastic demand is calculated using a demand elasticity of rice, κ, and formulated as:

6 316 H. Li, M.C.J. Bliemer, P.H.L.Bovy c ( k, θ ) c ( k,0) Q ( k) = Qo ( k) 1 κ, c ( k,0) (4) where c ( k, θ ) and c ( k,0) are the generalized OD costs from r to s dearting at time k with and without tolls, resectively. In this aer the generalized OD cost is calculated by the route flow weighted average route costs c ( k ) for each route P. That is, c q ( k) c ( k) P ( k) =, q ( k) P (5) where q ( k ) is the route flow rate (veh/h) of route from r to s dearting at time k. Before defining the route costs, we look at link cost formulation. With tolls in the network, link cost ca () t of entering link a at time t is comosed of the link travel time τ a () t and a link toll cost θ a () t when entering this link: c () t = ατ () t + θ (), t (6) a a a where α ( /h) is the value of time (VOT). Assuming additive link toll costs, the generalized route costs can be comuted as: c ( k) δ ( k, t) c ( t), = a a a t (7) where δ ( kt, ) is the dynamic link-route incidence indicator, which a equals one if flow dearting at time k from r to s following route ente link a at time t, and is zero otherwise. This indicator can also be used to relate the route flow rates q ( k ) to the link flow rates q (), t namely: a

7 Network reliability-based 317 q () t δ ( k,) t q ( k). = a a (, ) P k (8) The random utility function for an alternative route is the sum of observed (dis)utility, consisting of the generalized route costs c ( k ), and an unobserved term ε, exressed as: U ( k) = λ c ( k) + ε ( k). (9) where in our case λ = 1. The error term has zero mean and a variance σ. 2 ε The stochastic dynamic user-equilibrium assignment roblem can be formulated as the following equivalent variational inequality (VI) roblem (see e.g. He, 1997) where we would like to find equilibrium route flow rates q% ( k) such that: % ( % ) (, ) P k F ( k) q ( k) q ( k) 0, q ( k) Ω, (10) where c% ( k) F% ( k) = ( q% ( k) P ( k) Q ( k) ), (11) q ( k) and where P ( ) k is the roortion of travele choosing route among all route alternatives for an OD air (r, s). It can be calculated by any route choice models such as MNL, ath-size logit model, C-logit model. c% ( k) is the equilibrium route cost. Ω is the set of feasible route flow rates defined by flow conservation and nonnegativity constraints, resectively: P ( ) q ( k) = Q ( k), r, s RS, k K, (12)

8 318 H. Li, M.C.J. Bliemer, P.H.L.Bovy ( ) q ( k) 0, r, s RS, P, k K. (13) It is imortant to note that c ( k ), P ( k ), and Q ( k ) deend on the link toll levels θ, such that also the equilibrium route flow rates q% ( k) deend on the toll levels. Formulation of Network Travel Time Reliability We adot network travel time reliability as one of the objectives of the road authority for the toll system design. This network measure derives from tri and route based travel time reliability measures. The travel times constitute a distribution of a longer series of day-to-day fluctuations in travel time of tris made during a articular eriod of the day (e.g. morning eak). Route-based network travel time reliability is defined to exress the quality of service that a transort network rovides to its use over a longer stretch of time. On the route level, travel time is a stochastic variable following a class of distributions, due to stochastic caacities and stochastic travel demand. Travel time reliability is useful to evaluate network erformance under normal daily oerations. Different measures have been roosed as a travel time reliability indicator: robabilistic measures (Asakura and Kashiwadani, 1991; Asakura, 1998; Du and Nicholson, 1997; Bell et al., 1999; Yang et al., 2000; Sumalee et al., 2006), travel time standard deviation (Lomax et al., 1997; RAND Euroe and AVV, 2005), travel time skew (Van Lint et al., 2004), coefficient of variance, and the like. Standard deviation is chosen as an indicator of route travel time reliability, since it is a good measure to rovide the range of travel time variations and mathematically derivable. Although the day-to-day travel time distribution is tyically asymmetric, the effects of negative and ositive deviations of travel time can be formulated as a function of travel time standard deviation (RAND Euroe and AVV, 2005). This rovides some justification for the widesread use of standard deviation as an indicator of travel time reliability. The smaller the standard deviation is, the more reliable the route travel time is. The unreliability ˆ ρ of travel time valid for time interval ˆK K on route from origin r to destination s is defined as:

9 Network reliability-based 319 ˆ ρ ( ˆ τ ) = Var, (14) where ˆ τ is the average route travel time for route from r to s over eriod K ˆ. Thus, given link tolls θ, stochastic day-to-day link travel times τ a () t follow, from which the stochastic day-to-day route travel times τ ( k) and the stochastic average route travel times ˆ τ can be determined, yielding route travel time unreliability ˆ ρ. Travel time variability is caused by day-to-day caacity variations and travel demand fluctuations, and denotes the variability of average travel times at the same time eriod ˆK over the days. Within-day caacities are assumed constant during the time eriod of each day. Tri travel time unreliability is a unreliability indicator for a secific OD air which is imortant for use when making a tri from one lace to another. Tri travel time unreliability ˆ ρ during time interval ˆK for a traveler is defined on the basis of the corresonding route travel time unreliabilities of all used routes P and using average route flow rates as weights: ˆ ρ 1 = Eq ( ˆ ) ˆ, with ˆ ρ Q = Q ( k), ˆ Q P k Kˆ (15) where E( q ˆ ) is the exected average route flow rate during time eriod ˆK on route from r to s over a series of days. Network travel time unreliability is a useful measure to evaluate the road network erformance as a whole. It should take into account all OD ai travel time unreliabilities. Based on tri unreliability, network travel time unreliability ˆρ for eriod ˆK can be formulated as: ˆ ˆ Eq ( ˆ ) ˆ ρ Q ρ (, ) (, ) P ˆ = =. ρ Q ˆ (, ) (, ) Q ˆ (16)

10 320 H. Li, M.C.J. Bliemer, P.H.L.Bovy Since ˆ ρ deends on the link tolls, as exlained above, also ˆρ deends on the link tolls. This definition rovides an indicator to assess network reliability with different toll schemes, showing the global average travel time deviations for a secific day time eriod for an arbitrary user of this road network. For a secific network, this network unreliability measure can be used to comare road network reliability erformances under different conditions, such as with varying link caacities, with varying demand levels, with different tolling schemes, and so forth. For a comarison of network reliability between two different networks, the average network travel time needs to be taken into account in the definition of reliability. This average network travel time indicator ˆ, τ which again imlicitly deends on the link tolls, is also an imortant indicator of transort network services, which is calculated by (, ) P (, ) Eq ( ˆ ) ˆ τ ˆ τ =. ˆ Q (17) These two measures, ˆ τ and ˆ, ρ will be used in combination in the objective function of the otimal toll design roblem. The comlete reliability-based otimal toll design roblem, with the uer level objective derived here and the lower level equilibrium defined in the revious subsection, will be formulated in the next subsection. Formulation of Network Reliability-Based Otimal Toll Design Problem Based on the formulated lower level equilibrium, the formulation of network travel time (un)reliability, and the formulation of network travel time, the reliability-based otimal toll design roblem can be formulated as: γ min Z = ˆ τ + ˆ ρ = θ α (, ) P ( ˆ ( ))( ˆ ( ) γ ˆ ( ) θ τ θ + α ρ θ ) E q ˆ Q ( θ ) (18) (, )

11 Network reliability-based 321 subject to θ () t a 0, a A, t T, (19) % ( % ) (, ) P k F ( k) q ( k) q ( k) 0, q ( k) Ω. (20) where Ω is the set of feasible route flow rates defined by constraints (12) and (13). In the objective function in exression (18), α and γ are value of time (VOT) and value of reliability (VOR), resectively. In (18) it was found that the VOR increases or decreases accordingly to the increase or decrease of the VOT for different grous of eole, while the reliability ratio γ / α is aroximately constant (RAND Euroe and AVV, 2005). Exression (18) defines the objective function from a road authority s oint of view aiming to minimize network cost, which is a sum of network travel time and network unreliability. The network cost is a function of toll vector θ which we aim to derive, the stochastic caacity vector, and the elastic and stochastic OD demand. Thus, the reliability-based otimal toll design roblem roosed in this aer is a comlex roblem. It should be mentioned that the model roosed in this aer can be extended to include different user classes. Solution Aroach The comlexity of the design roblem defined above is NP-hard. In the work of Jeroslaw (1985) it is already mentioned that bi-level roblems are in general NP-hard. Due to the stochastic caacity and the fluctuating travel demand, we use a quasi-monte Carlo aroach (based on Halton draws; see Halton, 1960) to comute the unreliability measures in our design roblem. The framework of the solution rocess is illustrated in Figure 1. The design rocess can be generally divided into two arts: elastic demand calculation and otimal toll design. Using the base OD demand, initial network characteristics, and link tolls as inut, a dynamic traffic assignment is erformed. Based on the oututs from the assignment, tri travel costs can be derived, thus the temoral elastic demand can be derived using the demand function (see Equation (4)). However, every time with a new generated OD demand, a comlete new traffic assignment is erformed, leading to new costs and

12 322 H. Li, M.C.J. Bliemer, P.H.L.Bovy Initialization Base OD demand Mean link caacity Toll level θ Elastic demand ste Dynamic assignment Flow attern, travel times, tri costs Iteration+1 Demand function Temoral elastic demand Convergence criteria Elastic demand corresonding to toll θ Elastic demand vector Q θ ( ) Toll level θ Network Randomization in link caacities & OD demand Otimize toll θ Varying tolls levels Dynamic assignment Travel times & flows n times Network reliability corresonding to toll θ Otimal toll vector θ * Figure 1. Iterative toll otimization rocedure new elastic demand, until it converges. An iterative rocess is needed to calculate the elastic demand in terms of a toll vector θ.

13 Network reliability-based 323 In the next ste, with the derived elastic demand (for various toll vecto θ ) and initial network characteristics, link caacities and OD demand are randomized N times simultaneously (which models the travel time fluctuations from day-to-day) to determine the network reliability with resect to a toll vector θ. Different tolling levels can be alied and an otimal toll vector θ * will be determined based on the network cost minimization rincile using a simle grid search rocedure. Grid search rocedure will be inefficient for larger networks with many tolled links. More heuristic algorithms are needed for large networks with more comlicated toll schemes. Exeriment Results As a reliminary analysis, the roosed reliability-based otimal toll design model is alied to a small hyothetical five-link network (see Figure 2) with a single OD air (,). Since road ricing yields sare caacity on the charged link and the routes it belongs to, the charged routes can handle demand fluctuations more easily. However due to route flow shifts, the situation on other routes will become woe. Thus, on a network level, the network reliability is uncertain with tolls and needs to be analyzed. A single-link charging system is alied, in which only link 5 is tolled using a uniform toll for the whole time eriod. Links that will not be tolled receive a toll value of zero. link 1 link 4 route 2 r link 3 s r route 1 s link 2 link 5 (tolled) route 3 Figure 2. Five-link network Table 1 lists free flow travel times and other characteristics of the five links, such as caacities and critical and maximum seeds for the five-link network. The INDY dynamic traffic assignment model (see

14 324 H. Li, M.C.J. Bliemer, P.H.L.Bovy Bliemer et al., 2004; Bliemer, 2005) is used for comuting a stochastic dynamic user-equilibrium. Adated Smulde seed-density function is utilized in INDY for calculating travel time. Since in this study, we mainly focus on the analysis of the otential influence of road ricing on the network erformance, to investigate how the imacts can be after imlementing tolls in the network, fitly the multinomial logit (MNL) model is chosen for simlicity to model the choice behavior of travele. All three routes are aroximately equally attractive to the use when there are low flows through the routes, as the free flow route travel times are 3.5, 3.6, and 3.6, resectively for routes 1, 2, and 3. Only one time eriod of one hour is modeled and analyzed in which the base OD travel demand is 5000 veh/h. Dynamic assignment is erformed with uniform demand given every 10 minutes for 6 time intervals. While reliability and network erformance are analyzed for only one time eriod of one hour due to all the erformance indicato are average measures. 100 random Halton draws are erformed for modeling stochastic caacities and travel demand resectively. According to the caacities listed in Table 1, the network oerates around caacity. Since the traffic situations is much more unstable when the network is oerating around caacity, it is of more interests to look into the case when the network oerates around caacity, than the case with demand far less than caacity with reliable travel times. The adoted coefficients of day-to-day variation for demand and caacity are β = 0.2 and ω = 0.1, resectively, taken from Tu (2008) and AVV (1999). The demand elasticity to rice is assumed to be κ = 0.2. Table 1. Link Characteristics Free-flow travel time (min.) Link 1 Link 2 Link 3 Link 4 Link Caacity (veh/h) 2,500 2,500 2,800 2,500 2,500 Maximum seed (km/h) Seed at caacity (km/h) Figure 3 shows elastic demands in the modeled one-hour in relation to charge levels. As exected, the OD demand decreases as the toll

15 Network reliability-based 325 increases, because the generalized cost between the origin and destination increases due to increased toll costs on routes 1 and 3 and increased travel times due to route shifts towards route 2. Stochastic simulation results with elastic demand considering caacity variations and demand fluctuations simultaneously are resented for different toll levels. A reliability ratio of γ / α = 1.3 (see Exression (18)) is used in this case study, taken from Tseng et al. (2004). OD demand Charge on link 5 (euros) Figure 3. Elastic OD demand Figure 4 shows total network costs (see dotted line) defined in objective function (18) as the sum of network travel time and weighted network unreliability. The dashed line indicates average travel time er vehicle (equaling average total travel time divided by the elastic demand for each toll level) in terms of varied toll levels on link 5, from which it can be seen that although the OD demand declines with increasing toll levels, the average travel time er vehicle (see Equation (17)) kees nearly constant within the charge range of euros on link 5, while network unreliability decreases greatly with increasing toll levels on link 5 within the charge range of euros, as shown by the solid line. An otimal toll level of θ * 5 = 2.5 is derived for this single-link charging system with which network cost Z is minimized and coincidently network reliability is otimized as well.

16 326 H. Li, M.C.J. Bliemer, P.H.L.Bovy Objective function (minutes) network cost travel time comonent network unreliability comonent Charge on link 5 (euros) Figure 4. Objective network erformance The reason that network reliability can be imroved by tolls is due to decreased OD demand and less frequent route flow switches with increasing charge levels on link 5. With low charge levels, decreased OD demand has a stronger ositive influence on network reliability than route flow shifts do. With charge levels higher than 2.5 euros, most travele shift to uncharged route 2, leading to traffic flows on route 2 reaching its caacity. The travel time on route 2 increases significantly. Small changes in caacities lead to great changes in travel time, thus leads to higher travel time unreliability. As seen in Figure 4, network unreliability increases with charge levels higher than 2.5 euros. * Since the objective is to find the otimal toll level θ with which the network reliability is imroved, it is imortant to investigate the imacts of the otimal toll scheme on network travel time reliability. Figure 5 shows for comarison the route travel time distributions without tolls and with the otimal toll level of θ * = 2.5. Without tolls, route travel times distribute widely with large standard deviations, as shown in Figure 5(a), * while with the otimal toll vectorθ the route travel time distributions are very centralized with much smaller variations (see Figure 5(b)). Travel time reliability has been imroved greatly by charging a toll on link 5.

17 Network reliability-based ath one (tolled) 40 ath one (tolled) Probability ath two (non-tolled) ath three (tolled) Path travel time (minutes) (a) without tolls Probability ath two (non-tolled) ath three (tolled) Path travel time (minutes) (b) with tolls Figure 5. Route travel time distributions With resect to the exected route travel times, we notice an increase on route 1 and route 2, while a dro on route three after tolling. Travele on charged route 3 save considerable travel time and get more reliable travel times by aying the tolls. Travele on the other two routes take longer (comaring the exected travel time, see the quantified route travel times in Table 2), but face more reliable route travel times. Although route 1 is also charged, its travel time still increases a bit due to the fact that route 1 has an overlaing link with congested route 2. Based on above analyses, road ricing indeed may imrove network travel time reliability. Thus the otimal tolls can be designed from a network reliability oint of view. The roosed reliability-based otimal toll design model is romising at imroving network reliability, which is a crucial factor involved in the transort network design roblem nowadays. Note that the route costs (sum of travel time and tolls) with a toll of 2.5 euros are not equal for all routes (see Table 2). This does not violate the equilibrium rincile as we use a logit-based stochastic assignment instead of a deterministic equilibrium. Both caacity variations and demand fluctuations have significant influences on network reliability. Now let us quantitatively analyze the imacts of simultaneous fluctuations in both caacity and traffic demand, comared with the imacts from caacity or demand fluctuations searately. Table 2 illustrates mean route travel times, standard deviations and coefficients of variation (CoV) in terms of different affecting facto (see also Li, 2005). The bold cells highlight the results

18 328 H. Li, M.C.J. Bliemer, P.H.L.Bovy from simultaneous fluctuations in caacity and travel demand as analyzed in this aer. Table 2. Comarison Influence of Caacity Variations and Demand Fluctuations Charge Route travel Route travel Route Affecting facto ( ) time σ (min.) time μ (min.) CoV caacity variation % Route 1 demand fluctuation % simultaneous % caacity variation % 0 Route 2 demand fluctuation % simultaneous % caacity variation % Route 3 demand fluctuation % simultaneous % caacity variation % Route 1 demand fluctuation % simultaneous % caacity variation % 2.5 Route 2 demand fluctuation % simultaneous % caacity variation % Route 3 demand fluctuation % simultaneous % As exected, simultaneous fluctuations lead to larger variations in route travel times than either caacity or demand fluctuations searately do. Only modeling caacity or demand fluctuations and ignoring the other always underestimates the travel time variability. It can be noticed as well that simultaneous variations lead to shorter exected travel times, which can be exlained by the fact that the exectation of the minimum of a set of distributions is always smaller than or equal to the minimum of the exectation of each of the distributions (Bovy, 1984), since there are always oortunities to find a shorter travel time. Whereas without tolls, the CoV of route travel times caused from simultaneous fluctuations in both caacity and OD demand are larger

19 Network reliability-based 329 than 48%, with otimal tolls, variations of route travel times dramatically decline with much smaller CoV of route travel times. Tolls indeed can hel to imrove network travel time reliability due to decreased OD demand and less flow shifts after imlementing road ricing. Therefore, imroving network travel time reliability may be one of the imortant olicy objectives of road ricing. Conclusions In this aer, we roose network reliability as one of the romising olicy objectives of road ricing. A reliability-based otimal toll design model is roosed and formulated. Network erformance reliability is analyzed with resect to stochastic link caacities and OD demand with varied toll levels, which integrates reliability and uncertainty analysis, network equilibrium models, and Monte Carlo methods, to evaluate the erformance of a degradable road network with elastic and fluctuated demand. The roosed reliability-based otimal toll design model is alied to a simle network with a single-link charging system for a reliminary analysis. The otimal toll level can be determined with which the network cost is minimized and coincidently network reliability is also otimized in this case study. Imosing tolls can lead to decreased travel demand and less route choice shifts, which may have a ositive effect on the travel time reliabilities. Route travel times may increase, but will become more reliable. The roosed model can be utilized to otimize toll schemes by searching for otimal toll levels for all links and a set of otimal links for charging, aiming to imrove network travel time reliability and minimize network travel time. Imroving network reliability is one of the romising olicy objectives of road ricing, esecially in the context that reliability is becoming increasingly imortant in network design roblems. Although only a simle network is tested with some assumtions, the roosed model can be alied for a general otimal toll design roblem aiming to imrove network travel time reliability. Furthermore, it can be extended by including dearture time choice or to the design of time-varying toll systems. Further research can be done with alications on realistic networks to testify the roosed model.

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